Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508078 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #23917 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1049 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 1049 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,22)(13,26)(14,29)(15,18)(17,20)(19,31)(21,37)(23,36)(24,40)(25,32)(27,35)(28,33)(30,34)(38,39);(2,5)(6,8)(9,11,15,13)(10,16,18,26)(12,23,25,39)(14,28,31,21)(17,34,35,40)(19,37,29,33)(20,30,27,24)(22,36,32,38);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,20)(13,27)(14,30)(15,25)(16,32)(18,35)(19,38)(21,39)(22,26)(23,28)(24,31)(29,36)(33,40)(34,37);(3,7)(6,8)(10,18)(16,26)(17,35)(19,29)(22,32)(33,37)(34,40)(36,38)Polycyclic presentation:
2^9|b^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415275);